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Commutative idempotent groupoids and the constraint satisfaction problem

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Abstract

A restatement of the Algebraic Dichotomy Conjecture, due to Maroti and McKenzie, postulates that if a finite algebra A possesses a weak near-unanimity term, then the corresponding constraint satisfaction problem is tractable. A binary operation is weak near-unanimity if and only if it is both commutative and idempotent. Thus if the dichotomy conjecture is true, any finite commutative, idempotent groupoid (CI groupoid) will be tractable. It is known that every semilattice (i.e., an associative CI groupoid) is tractable. A groupoid identity is of Bol–Moufang type if the same three variables appear on either side, one of the variables is repeated, the remaining two variables appear once, and the variables appear in the same order on either side (for example, \({x(x(yz)) \approx (x(xy))z}\)). These identities can be thought of as generalizations of associativity. We show that there are exactly 8 varieties of CI groupoids defined by a single additional identity of Bol–Moufang type, derive some of their important structural properties, and use that structure theory to show that 7 of the varieties are tractable. We also characterize the finite members of the variety of CI groupoids satisfying the self-distributive law \({x(yz) \approx (xy)(xz)}\), and show that they are tractable.

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References

  1. Barto, L., Kozik, M.: Constraint satisfaction problems of bounded width. In: 2009 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009), pp. 595–603. IEEE Computer Soc., Los Alamitos, CA (2009)

  2. Bergman, C.: Universal Algebra: Fundamentals and Selected Topics. CRC Press, Boca Raton, FL (2012)

  3. Bulatov, A., Dalmau, V.: A simple algorithm for Mal′tsev constraints. SIAM J. Comput. 36, 16–27 (electronic) (2006)

  4. Bulatov, A., Jeavons, P.: Algebraic structures in combinatorial problems. Tech. rep., Technische Universitat Dresden (2001). MATH–AL–4–2001, http://www.cs.sfu.ca/~abulatov/papers/varieties.ps

  5. Bulatov, A., Jeavons, P., Krokhin, A.: Classifying the complexity of constraints using finite algebras. SIAM J. Comput. 34, 720–742 (electronic) (2005)

  6. Bulatov, A.A.: Combinatorial problems raised from 2-semilattices. J. Algebra 298, 321–339 (2006)

  7. Bulatov, A.A., Valeriote, M.: Recent results on the algebraic approach to the CSP. In: N. Creignou, P.G. Kolaitis, H. Vollmer (eds.) Complexity of Constraints, pp. 68–92. Springer, Berlin (2008)

  8. Burris, S., Sankappanavar, H.P.: A course in universal algebra, Graduate Texts in Mathematics, vol. 78. Springer, New York (1981)

  9. Dalmau, V.: Generalized majority-minority operations are tractable. Log. Methods Comput. Sci. 2, 4:1, 14 pp. (2006)

  10. Feder, T., Vardi, M.Y.: The computational structure of monotone monadic SNP and constraint satisfaction: a study through Datalog and group theory. SIAM J. Comput. 28, 57–104 (electronic) (1999)

  11. Fenyves, F.: Extra loops. II. On loops with identities of Bol-Moufang type. Publ. Math. Debrecen 16, 187–192 (1969)

  12. Freese, R., Kiss, E., Valeriote, M.: Universal Algebra Calculator (2011). Available at: www.uacalc.org

  13. Idziak, P., Marković, P., McKenzie, R., Valeriote, M., Willard, R.: Tractability and learnability arising from algebras with few subpowers. SIAM J. Comput. 39, 3023–3037 (2010)

  14. Jeavons, P.: On the algebraic structure of combinatorial problems. Theoret. Comput. Sci. 200, 185–204 (1998)

  15. Jeavons, P., Cohen, D., Gyssens, M.: Closure properties of constraints. J. ACM 44, 527–548 (1997)

  16. Ježek, J., Kepka, T.: The lattice of varieties of commutative abelian distributive groupoids. Algebra Universalis 5, 225–237 (1975)

  17. Ježek, J., Kepka, T., Němec, P.: Distributive groupoids. Rozpravy Československé Akad. Věd Řada Mat. Přírod. Věd 91, 94 (1981)

  18. Kepka, T.: Commutative distributive groupoids. Acta Univ. Carolin.—Math. Phys. 19(2), 45–58 (1978)

  19. Kepka, T., Němec, P.: Commutative Moufang loops and distributive groupoids of small orders. Czechoslovak Math. J. 31(106), 633–669 (1981)

  20. Kozik, M., Krokhin, A., Valeriote, M.A., Willard, R.: Characterizations of several Maltsev conditions. preprint (2013)

  21. Kunen, K.: Quasigroups, loops, and associative laws. J. Algebra 185, 194–204 (1996). DOI 10.1006/jabr.1996.0321. URL http://dx.doi.org/10.1006/jabr.1996.0321

  22. Mackworth, A.: Consistency in networks of relations. Artificial Intelligence 8(1), 99–118 (1977). Reprinted in Readings in Artificial Intelligence, B. L. Webber and N. J. Nilsson (eds.), Tioga Publ. Col., Palo Alto, CA, pp. 69–78, 1981.

  23. McCune, W.: Prover9 and mace4 (2005–2010). http://www.cs.unm.edu/~mccune/prover9/

  24. Mel′nik, I.I.: Normal closures of perfect varieties of universal algebras. In: Ordered sets and lattices, No. 1 (Russian), pp. 56–65. Izdat. Saratov. Univ., Saratov (1971)

  25. Phillips, J.D., Vojtěchovský, P.: The varieties of loops of Bol-Moufang type. Algebra Universalis 54, 259–271 (2005)

  26. Phillips, J.D., Vojtěchovský, P.: The varieties of quasigroups of Bol-Moufang type: an equational reasoning approach. J. Algebra 293, 17–33 (2005)

  27. Płonka, J.: On a method of construction of abstract algebras. Fund. Math. 61, 183–189 (1967)

  28. Płonka, J.: On equational classes of abstract algebras defined by regular equations. Fund. Math. 64, 241–247 (1969)

  29. Quackenbush, R.W.: Varieties of Steiner loops and Steiner quasigroups. Canad. J. Math. 28, 1187–1198 (1976)

  30. Romanowska, A.: On regular and regularised varieties. Algebra Universalis 23, 215–241 (1986)

  31. Romanowska, A.B., Smith, J.D.H.: Modes. World Scientific Publishing Co. Inc., River Edge, NJ (2002)

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Correspondence to David Failing.

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Presented by M. Maroti.

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Bergman, C., Failing, D. Commutative idempotent groupoids and the constraint satisfaction problem. Algebra Univers. 73, 391–417 (2015). https://doi.org/10.1007/s00012-015-0323-6

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